A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
2.8. The Pease–Braginskii current and radiative collapse
Over forty years ago Pease [8] and Braginskii [9] independently predicted that a pure Z-pinch
pressure equilibrium would have a unique critical current I PB when the Joule heating balances
bremsstrahlung radiation loss. For a hydrogen plasma and for a parabolic density profile and
uniform current I PB is 0.433 (ln ) 1/2 MA. How this arises can easily be shown from simple
arguments. The bremsstrahlung loss rate per unit volume is β b Zn 2 e T 1/2 where β b is a constant
equal to 1.69 × 10 −38 Wm 3 (eV) −1/2 . The Joule heating per unit volume is η ⊥ J 2 where the
cross-field or transverse Spitzer resistivity η ⊥ is 1.03 × 10 −4 ZlnT −3/2 = (αT 3/2 ) −1 m.
On balancing the Joule heating against the radiation loss
η ⊥ J 2 ∼ = βb Zn 2 e T 1/2 (2.63)
the temperature dependences of the two phenomena combine to give J ∝ p = n i k B T(1+Z).
However, pressure balance for a volume distributed current is approximately JB ∼ = p/a where
a is the pinch radius. Therefore it follows that Ba is approximately constant. Ba is just µ o I/2π
and here demonstrates that there is a unique critical current. For a parabolic density profile
associated with uniform current density and temperature and one or several ion species the
critical current is
I PB = 8√ ( )
3k B 1+Z
1+〈Z〉
√
µ 0 αβb 2Z
= 0.433(ln )1/2 MA. (2.64)
2〈Z 2 〉 1/2
This is a generalization of equation (2.36) for arbitrary Z showing that for pure bremsstrahlung
(no line radiation or opacity effects) the maximum effect of Z>1 in a single species plasma is
to halve the Pease–Braginskii current. However, for an optimum doping of a Z = 1 plasma with
a fraction 2Z/(Z 2 − 1) of Z 3 ions the factor (1+〈Z〉)/2〈Z 2 〉 1/2 is √ (2Z +1) 1/2 /(Z +1)
and can be much less than 0.5.
Pease pointed out that such a current should depend only on fundamental constants, and
I A F(α fs ) where I A is the Alfvén–Lawson current (see section 2.2) given in terms of r e , the
classical radius of the electron defined by
e 2
r e =
(2.65)
4πε 0 mc 2
such that
I A = ec . (2.66)
r e
The fine structure constant is given by
α fs =
e2
4πε ∼ = 1
(2.67)
0¯hc 137
and F(α fs ) is proportional to α −1/2
fs
. Pereira [116] developed these ideas further showing the
relationship of the analysis to fundamental atomic parameters and ln . The Alfvén–Lawson
current (see equation (2.36)) is related to the maximum electron current that can propagate as
a beam in a stationary ion background, i.e. all electrons originate from ‘the cathode’ and no
return electron flow or even E r /B θ electron guiding-centre flow are allowed. This current is
entirely a singular electron current (equation (2.19)) where now P ⊥e is n e m e ve 2 and J z is n e ev e .
Thus, for relativistic electrons where m = γm e , γ = (1 − β 2 ) −1/2 and β = v e /c the current is
I A = 17 000γβ A. It could be, but has not yet been proven, that this is the maximum runaway
electron current that can occur in a disruption (see section 3.14).
The first calculation of radiation collapse assumed constant current and temperature [117].
With the current fixed above the Pease–Braginskii value indefinite collapse would occur until
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